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Theorem gsumhashmul 31898
Description: Express a group sum by grouping by nonzero values. (Contributed by Thierry Arnoux, 22-Jun-2024.)
Hypotheses
Ref Expression
gsumhashmul.b 𝐵 = (Base‘𝐺)
gsumhashmul.z 0 = (0g𝐺)
gsumhashmul.x · = (.g𝐺)
gsumhashmul.g (𝜑𝐺 ∈ CMnd)
gsumhashmul.f (𝜑𝐹:𝐴𝐵)
gsumhashmul.1 (𝜑𝐹 finSupp 0 )
Assertion
Ref Expression
gsumhashmul (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
Distinct variable groups:   𝑥, 0   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥
Allowed substitution hint:   · (𝑥)

Proof of Theorem gsumhashmul
Dummy variables 𝑡 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumhashmul.f . . . . . . 7 (𝜑𝐹:𝐴𝐵)
2 suppssdm 8108 . . . . . . . 8 (𝐹 supp 0 ) ⊆ dom 𝐹
32, 1fssdm 6688 . . . . . . 7 (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
41, 3feqresmpt 6911 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥)))
54oveq2d 7373 . . . . 5 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥))))
6 gsumhashmul.b . . . . . 6 𝐵 = (Base‘𝐺)
7 gsumhashmul.z . . . . . 6 0 = (0g𝐺)
8 gsumhashmul.g . . . . . 6 (𝜑𝐺 ∈ CMnd)
9 gsumhashmul.1 . . . . . . . 8 (𝜑𝐹 finSupp 0 )
10 relfsupp 9307 . . . . . . . . 9 Rel finSupp
1110brrelex1i 5688 . . . . . . . 8 (𝐹 finSupp 0𝐹 ∈ V)
129, 11syl 17 . . . . . . 7 (𝜑𝐹 ∈ V)
131ffnd 6669 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
1412, 13fndmexd 7843 . . . . . 6 (𝜑𝐴 ∈ V)
15 ssidd 3967 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
166, 7, 8, 14, 1, 15, 9gsumres 19690 . . . . 5 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg 𝐹))
17 nfcv 2907 . . . . . 6 𝑥(𝐹‘(1st𝑧))
18 fveq2 6842 . . . . . 6 (𝑥 = (1st𝑧) → (𝐹𝑥) = (𝐹‘(1st𝑧)))
199fsuppimpd 9312 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
20 ssidd 3967 . . . . . 6 (𝜑𝐵𝐵)
211adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝐹:𝐴𝐵)
223sselda 3944 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥𝐴)
2321, 22ffvelcdmd 7036 . . . . . 6 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹𝑥) ∈ 𝐵)
241ffund 6672 . . . . . . . . 9 (𝜑 → Fun 𝐹)
25 funrel 6518 . . . . . . . . 9 (Fun 𝐹 → Rel 𝐹)
26 reldif 5771 . . . . . . . . 9 (Rel 𝐹 → Rel (𝐹 ∖ (V × { 0 })))
2724, 25, 263syl 18 . . . . . . . 8 (𝜑 → Rel (𝐹 ∖ (V × { 0 })))
28 1stdm 7972 . . . . . . . 8 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
2927, 28sylan 580 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
307fvexi 6856 . . . . . . . . . . . 12 0 ∈ V
3130a1i 11 . . . . . . . . . . 11 (𝜑0 ∈ V)
32 fressupp 31603 . . . . . . . . . . 11 ((Fun 𝐹𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
3324, 12, 31, 32syl3anc 1371 . . . . . . . . . 10 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
3433dmeqd 5861 . . . . . . . . 9 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = dom (𝐹 ∖ (V × { 0 })))
352a1i 11 . . . . . . . . . 10 (𝜑 → (𝐹 supp 0 ) ⊆ dom 𝐹)
36 ssdmres 5960 . . . . . . . . . 10 ((𝐹 supp 0 ) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
3735, 36sylib 217 . . . . . . . . 9 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
3834, 37eqtr3d 2778 . . . . . . . 8 (𝜑 → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
3938adantr 481 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
4029, 39eleqtrd 2840 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st𝑧) ∈ (𝐹 supp 0 ))
4124funresd 6544 . . . . . . . . . . 11 (𝜑 → Fun (𝐹 ↾ (𝐹 supp 0 )))
4241adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → Fun (𝐹 ↾ (𝐹 supp 0 )))
4337eleq2d 2823 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (𝐹 supp 0 )))
4443biimpar 478 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
45 simpr 485 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ (𝐹 supp 0 ))
4645fvresd 6862 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥))
47 funopfvb 6898 . . . . . . . . . . 11 ((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 ))))
4847biimpa 477 . . . . . . . . . 10 (((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥)) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
4942, 44, 46, 48syl21anc 836 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
5033adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
5149, 50eleqtrd 2840 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹𝑥)⟩ ∈ (𝐹 ∖ (V × { 0 })))
52 eqeq2 2748 . . . . . . . . . . 11 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → (𝑧 = 𝑣𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
5352bibi2d 342 . . . . . . . . . 10 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → ((𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ (𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
5453ralbidv 3174 . . . . . . . . 9 (𝑣 = ⟨𝑥, (𝐹𝑥)⟩ → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
5554adantl 482 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑣 = ⟨𝑥, (𝐹𝑥)⟩) → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)))
56 fvexd 6857 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (2nd𝑧) ∈ V)
5727ad3antrrr 728 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → Rel (𝐹 ∖ (V × { 0 })))
58 simplr 767 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
59 1st2nd 7971 . . . . . . . . . . . . . . . . 17 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
6057, 58, 59syl2anc 584 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
61 opeq1 4830 . . . . . . . . . . . . . . . . 17 (𝑥 = (1st𝑧) → ⟨𝑥, (2nd𝑧)⟩ = ⟨(1st𝑧), (2nd𝑧)⟩)
6261adantl 482 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ⟨𝑥, (2nd𝑧)⟩ = ⟨(1st𝑧), (2nd𝑧)⟩)
6360, 62eqtr4d 2779 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨𝑥, (2nd𝑧)⟩)
64 difssd 4092 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ∖ (V × { 0 })) ⊆ 𝐹)
6564sselda 3944 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧𝐹)
6665adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧𝐹)
6763, 66eqeltrrd 2839 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹)
6863, 67jca 512 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (𝑧 = ⟨𝑥, (2nd𝑧)⟩ ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹))
69 opeq2 4831 . . . . . . . . . . . . . . . 16 (𝑦 = (2nd𝑧) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, (2nd𝑧)⟩)
7069eqeq2d 2747 . . . . . . . . . . . . . . 15 (𝑦 = (2nd𝑧) → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, (2nd𝑧)⟩))
7169eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑦 = (2nd𝑧) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹))
7270, 71anbi12d 631 . . . . . . . . . . . . . 14 (𝑦 = (2nd𝑧) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (𝑧 = ⟨𝑥, (2nd𝑧)⟩ ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐹)))
7356, 68, 72spcedv 3557 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
74 vex 3449 . . . . . . . . . . . . . 14 𝑥 ∈ V
7574elsnres 5977 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐹 ↾ {𝑥}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
7673, 75sylibr 233 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ (𝐹 ↾ {𝑥}))
7713ad3antrrr 728 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝐹 Fn 𝐴)
7822ad2antrr 724 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑥𝐴)
79 fnressn 7104 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
8077, 78, 79syl2anc 584 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
8176, 80eleqtrd 2840 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 ∈ {⟨𝑥, (𝐹𝑥)⟩})
82 elsni 4603 . . . . . . . . . . 11 (𝑧 ∈ {⟨𝑥, (𝐹𝑥)⟩} → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
8381, 82syl 17 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st𝑧)) → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
84 simpr 485 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑧 = ⟨𝑥, (𝐹𝑥)⟩)
8584fveq2d 6846 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → (1st𝑧) = (1st ‘⟨𝑥, (𝐹𝑥)⟩))
86 fvex 6855 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
8774, 86op1st 7929 . . . . . . . . . . 11 (1st ‘⟨𝑥, (𝐹𝑥)⟩) = 𝑥
8885, 87eqtr2di 2793 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑥 = (1st𝑧))
8983, 88impbida 799 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
9089ralrimiva 3143 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹𝑥)⟩))
9151, 55, 90rspcedvd 3583 . . . . . . 7 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣))
92 reu6 3684 . . . . . . 7 (∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st𝑧) ↔ ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st𝑧) ↔ 𝑧 = 𝑣))
9391, 92sylibr 233 . . . . . 6 ((𝜑𝑥 ∈ (𝐹 supp 0 )) → ∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st𝑧))
9417, 6, 7, 18, 8, 19, 20, 23, 40, 93gsummptf1o 19740 . . . . 5 (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑥))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))))
955, 16, 943eqtr3d 2784 . . . 4 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))))
96 simpr 485 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
9796eldifad 3922 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧𝐹)
98 funfv1st2nd 7978 . . . . . . 7 ((Fun 𝐹𝑧𝐹) → (𝐹‘(1st𝑧)) = (2nd𝑧))
9924, 97, 98syl2an2r 683 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝐹‘(1st𝑧)) = (2nd𝑧))
10099mpteq2dva 5205 . . . . 5 (𝜑 → (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧))) = (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧)))
101100oveq2d 7373 . . . 4 (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st𝑧)))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))))
10295, 101eqtrd 2776 . . 3 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))))
103 nfcv 2907 . . . 4 𝑧(1st𝑡)
104 fvex 6855 . . . . 5 (2nd𝑡) ∈ V
105 fvex 6855 . . . . 5 (1st𝑡) ∈ V
106104, 105op2ndd 7932 . . . 4 (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ → (2nd𝑧) = (1st𝑡))
107 resfnfinfin 9276 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝐹 supp 0 ) ∈ Fin) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
10813, 19, 107syl2anc 584 . . . . 5 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
10933, 108eqeltrrd 2839 . . . 4 (𝜑 → (𝐹 ∖ (V × { 0 })) ∈ Fin)
11033rneqd 5893 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = ran (𝐹 ∖ (V × { 0 })))
111 rnresss 5973 . . . . . 6 ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ ran 𝐹
1121frnd 6676 . . . . . 6 (𝜑 → ran 𝐹𝐵)
113111, 112sstrid 3955 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
114110, 113eqsstrrd 3983 . . . 4 (𝜑 → ran (𝐹 ∖ (V × { 0 })) ⊆ 𝐵)
115 2ndrn 7973 . . . . 5 ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
11627, 115sylan 580 . . . 4 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
117 relcnv 6056 . . . . . . . 8 Rel 𝐹
118 reldif 5771 . . . . . . . 8 (Rel 𝐹 → Rel (𝐹 ∖ ({ 0 } × V)))
119117, 118mp1i 13 . . . . . . 7 (𝜑 → Rel (𝐹 ∖ ({ 0 } × V)))
120 1st2nd 7971 . . . . . . 7 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st𝑡), (2nd𝑡)⟩)
121119, 120sylan 580 . . . . . 6 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st𝑡), (2nd𝑡)⟩)
122 cnvdif 6096 . . . . . . . . . 10 (𝐹 ∖ (V × { 0 })) = (𝐹(V × { 0 }))
123 cnvxp 6109 . . . . . . . . . . 11 (V × { 0 }) = ({ 0 } × V)
124123difeq2i 4079 . . . . . . . . . 10 (𝐹(V × { 0 })) = (𝐹 ∖ ({ 0 } × V))
125122, 124eqtri 2764 . . . . . . . . 9 (𝐹 ∖ (V × { 0 })) = (𝐹 ∖ ({ 0 } × V))
126125eqimss2i 4003 . . . . . . . 8 (𝐹 ∖ ({ 0 } × V)) ⊆ (𝐹 ∖ (V × { 0 }))
127126a1i 11 . . . . . . 7 (𝜑 → (𝐹 ∖ ({ 0 } × V)) ⊆ (𝐹 ∖ (V × { 0 })))
128127sselda 3944 . . . . . 6 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → 𝑡(𝐹 ∖ (V × { 0 })))
129121, 128eqeltrrd 2839 . . . . 5 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
130105, 104opelcnv 5837 . . . . 5 (⟨(1st𝑡), (2nd𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })) ↔ ⟨(2nd𝑡), (1st𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
131129, 130sylib 217 . . . 4 ((𝜑𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → ⟨(2nd𝑡), (1st𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
13227adantr 481 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → Rel (𝐹 ∖ (V × { 0 })))
133 eqidd 2737 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} = {𝑧})
134 cnvf1olem 8042 . . . . . . . . 9 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ {𝑧} = {𝑧})) → ( {𝑧} ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑧 = { {𝑧}}))
135134simpld 495 . . . . . . . 8 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ {𝑧} = {𝑧})) → {𝑧} ∈ (𝐹 ∖ (V × { 0 })))
136132, 96, 133, 135syl12anc 835 . . . . . . 7 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} ∈ (𝐹 ∖ (V × { 0 })))
137136, 125eleqtrdi 2848 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → {𝑧} ∈ (𝐹 ∖ ({ 0 } × V)))
138 eqeq2 2748 . . . . . . . . 9 (𝑢 = {𝑧} → (𝑡 = 𝑢𝑡 = {𝑧}))
139138bibi2d 342 . . . . . . . 8 (𝑢 = {𝑧} → ((𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
140139ralbidv 3174 . . . . . . 7 (𝑢 = {𝑧} → (∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
141140adantl 482 . . . . . 6 (((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑢 = {𝑧}) → (∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧})))
142117, 118mp1i 13 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → Rel (𝐹 ∖ ({ 0 } × V)))
143 simplr 767 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 ∈ (𝐹 ∖ ({ 0 } × V)))
144 simpr 485 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
145 df-rel 5640 . . . . . . . . . . . . . 14 (Rel (𝐹 ∖ ({ 0 } × V)) ↔ (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
146119, 145sylib 217 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
147146ad3antrrr 728 . . . . . . . . . . . 12 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
148147, 143sseldd 3945 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 ∈ (V × V))
149 2nd1st 7970 . . . . . . . . . . 11 (𝑡 ∈ (V × V) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
150148, 149syl 17 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
151144, 150eqtr4d 2779 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑧 = {𝑡})
152 cnvf1olem 8042 . . . . . . . . . 10 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = {𝑡})) → (𝑧(𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 = {𝑧}))
153152simprd 496 . . . . . . . . 9 ((Rel (𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = {𝑡})) → 𝑡 = {𝑧})
154142, 143, 151, 153syl12anc 835 . . . . . . . 8 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩) → 𝑡 = {𝑧})
15527ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → Rel (𝐹 ∖ (V × { 0 })))
15696ad2antrr 724 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
157 simpr 485 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 = {𝑧})
158 cnvf1olem 8042 . . . . . . . . . . 11 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = {𝑧})) → (𝑡(𝐹 ∖ (V × { 0 })) ∧ 𝑧 = {𝑡}))
159158simprd 496 . . . . . . . . . 10 ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = {𝑧})) → 𝑧 = {𝑡})
160155, 156, 157, 159syl12anc 835 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 = {𝑡})
161146ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → (𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
162 simplr 767 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 ∈ (𝐹 ∖ ({ 0 } × V)))
163161, 162sseldd 3945 . . . . . . . . . 10 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑡 ∈ (V × V))
164163, 149syl 17 . . . . . . . . 9 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → {𝑡} = ⟨(2nd𝑡), (1st𝑡)⟩)
165160, 164eqtrd 2776 . . . . . . . 8 ((((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = {𝑧}) → 𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
166154, 165impbida 799 . . . . . . 7 (((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (𝐹 ∖ ({ 0 } × V))) → (𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧}))
167166ralrimiva 3143 . . . . . 6 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = {𝑧}))
168137, 141, 167rspcedvd 3583 . . . . 5 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃𝑢 ∈ (𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢))
169 reu6 3684 . . . . 5 (∃!𝑡 ∈ (𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ ∃𝑢 ∈ (𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩ ↔ 𝑡 = 𝑢))
170168, 169sylibr 233 . . . 4 ((𝜑𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃!𝑡 ∈ (𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd𝑡), (1st𝑡)⟩)
171103, 6, 7, 106, 8, 109, 114, 116, 131, 170gsummptf1o 19740 . . 3 (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd𝑧))) = (𝐺 Σg (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡))))
172 fveq2 6842 . . . . . 6 (𝑡 = 𝑧 → (1st𝑡) = (1st𝑧))
173172cbvmptv 5218 . . . . 5 (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡)) = (𝑧 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑧))
17433cnveqd 5831 . . . . . . 7 (𝜑(𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
175174, 125eqtr2di 2793 . . . . . 6 (𝜑 → (𝐹 ∖ ({ 0 } × V)) = (𝐹 ↾ (𝐹 supp 0 )))
176175mpteq1d 5200 . . . . 5 (𝜑 → (𝑧 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑧)) = (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧)))
177173, 176eqtrid 2788 . . . 4 (𝜑 → (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡)) = (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧)))
178177oveq2d 7373 . . 3 (𝜑 → (𝐺 Σg (𝑡 ∈ (𝐹 ∖ ({ 0 } × V)) ↦ (1st𝑡))) = (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))))
179102, 171, 1783eqtrd 2780 . 2 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))))
180 nfcv 2907 . . 3 𝑦(1st𝑧)
181 nfv 1917 . . 3 𝑥𝜑
182 vex 3449 . . . 4 𝑦 ∈ V
18374, 182op1std 7931 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
184 relcnv 6056 . . . 4 Rel (𝐹 ↾ (𝐹 supp 0 ))
185184a1i 11 . . 3 (𝜑 → Rel (𝐹 ↾ (𝐹 supp 0 )))
186 cnvfi 9124 . . . 4 ((𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
187108, 186syl 17 . . 3 (𝜑(𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
188112adantr 481 . . . 4 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → ran 𝐹𝐵)
189184a1i 11 . . . . . . 7 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → Rel (𝐹 ↾ (𝐹 supp 0 )))
190 simpr 485 . . . . . . 7 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → 𝑧(𝐹 ↾ (𝐹 supp 0 )))
191 1stdm 7972 . . . . . . 7 ((Rel (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
192189, 190, 191syl2anc 584 . . . . . 6 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
193 df-rn 5644 . . . . . 6 ran (𝐹 ↾ (𝐹 supp 0 )) = dom (𝐹 ↾ (𝐹 supp 0 ))
194192, 193eleqtrrdi 2849 . . . . 5 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ ran (𝐹 ↾ (𝐹 supp 0 )))
195111, 194sselid 3942 . . . 4 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ ran 𝐹)
196188, 195sseldd 3945 . . 3 ((𝜑𝑧(𝐹 ↾ (𝐹 supp 0 ))) → (1st𝑧) ∈ 𝐵)
197180, 181, 6, 183, 185, 187, 8, 196gsummpt2d 31891 . 2 (𝜑 → (𝐺 Σg (𝑧(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st𝑧))) = (𝐺 Σg (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))))
198 df-ima 5646 . . . . . . 7 (𝐹 “ (𝐹 supp 0 )) = ran (𝐹 ↾ (𝐹 supp 0 ))
199 supppreima 31606 . . . . . . . . 9 ((Fun 𝐹𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
20024, 12, 31, 199syl3anc 1371 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
201200imaeq2d 6013 . . . . . . 7 (𝜑 → (𝐹 “ (𝐹 supp 0 )) = (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))))
202198, 201eqtr3id 2790 . . . . . 6 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))))
203 funimacnv 6582 . . . . . . 7 (Fun 𝐹 → (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
20424, 203syl 17 . . . . . 6 (𝜑 → (𝐹 “ (𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
205 difssd 4092 . . . . . . 7 (𝜑 → (ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹)
206 df-ss 3927 . . . . . . 7 ((ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹 ↔ ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
207205, 206sylib 217 . . . . . 6 (𝜑 → ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
208202, 204, 2073eqtrd 2780 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
209193, 208eqtr3id 2790 . . . 4 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
2108cmnmndd 19586 . . . . . . 7 (𝜑𝐺 ∈ Mnd)
211210adantr 481 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
212108adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
213 imafi2 31628 . . . . . . 7 ((𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
214212, 186, 2133syl 18 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
215193, 113eqsstrrid 3993 . . . . . . 7 (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
216215sselda 3944 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝑥𝐵)
217 gsumhashmul.x . . . . . . 7 · = (.g𝐺)
2186, 217gsumconst 19711 . . . . . 6 ((𝐺 ∈ Mnd ∧ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin ∧ 𝑥𝐵) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
219211, 214, 216, 218syl3anc 1371 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
220 cnvresima 6182 . . . . . . . 8 ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) = ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 ))
221209eleq2d 2823 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (ran 𝐹 ∖ { 0 })))
222221biimpa 477 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → 𝑥 ∈ (ran 𝐹 ∖ { 0 }))
223222snssd 4769 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → {𝑥} ⊆ (ran 𝐹 ∖ { 0 }))
224 sspreima 7018 . . . . . . . . . . 11 ((Fun 𝐹 ∧ {𝑥} ⊆ (ran 𝐹 ∖ { 0 })) → (𝐹 “ {𝑥}) ⊆ (𝐹 “ (ran 𝐹 ∖ { 0 })))
22524, 223, 224syl2an2r 683 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) ⊆ (𝐹 “ (ran 𝐹 ∖ { 0 })))
226200adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 supp 0 ) = (𝐹 “ (ran 𝐹 ∖ { 0 })))
227225, 226sseqtrrd 3985 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ))
228 df-ss 3927 . . . . . . . . 9 ((𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ) ↔ ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (𝐹 “ {𝑥}))
229227, 228sylib 217 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (𝐹 “ {𝑥}))
230220, 229eqtr2id 2789 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 “ {𝑥}) = ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}))
231230fveq2d 6846 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (♯‘(𝐹 “ {𝑥})) = (♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})))
232231oveq1d 7372 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → ((♯‘(𝐹 “ {𝑥})) · 𝑥) = ((♯‘((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
233219, 232eqtr4d 2779 . . . 4 ((𝜑𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(𝐹 “ {𝑥})) · 𝑥))
234209, 233mpteq12dva 5194 . . 3 (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥))) = (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥)))
235234oveq2d 7373 . 2 (𝜑 → (𝐺 Σg (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ ((𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
236179, 197, 2353eqtrd 2780 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  ∃!wreu 3351  Vcvv 3445  cdif 3907  cin 3909  wss 3910  {csn 4586  cop 4592   cuni 4865   class class class wbr 5105  cmpt 5188   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Rel wrel 5638  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920   supp csupp 8092  Fincfn 8883   finSupp cfsupp 9305  chash 14230  Basecbs 17083  0gc0g 17321   Σg cgsu 17322  Mndcmnd 18556  .gcmg 18872  CMndccmn 19562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564
This theorem is referenced by:  elrspunidl  32203
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